Solve the exponential equation for $x$. 2 3 x − 4 64 x − 5 = 2 8 x − 1 \dfrac{2\^{3x-4}}{64\^{ x-5}}=2\^{ 8x-1} $x=$
Solution: The strategy Let's write $64$ in base $2$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $2$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 2 3 x − 4 64 x − 5 = 2 3 x − 4 ( 2 6 ) x − 5 = 2 3 x − 4 2 6 x − 30 = 2 3 x − 4 − ( 6 x − 30 ) = 2 − 3 x + 26 ( 64 = 2 6 ) ( ( a n ) m = a n ⋅ m ) ( a n a m = a n − m ) \begin{aligned}\dfrac{2\^{3x-4}}{64\^{ x-5}}&=\dfrac{2\^{3x-4}}{(2^6)\^{ x-5}}&&&&(64=2^6) \\\\\\\\ &=\dfrac{2\^{ C{3x-4}}}{2\^{ {6x-30}}} &&&&((a^n)^m=a^{n\cdot m})\\\\\\\\ &=2\^{ C{3x-4} \ - \ ({6x-30})}&&&&(\dfrac{a^n}{a^m}=a^{n-m})\\\\\\\\ &=2\^{ -3x+26} \end{aligned} Solving the equation We obtain the following equation. 2 − 3 x + 26 = 2 8 x − 1 2\^{-3x+26}=2\^{ 8x-1} Now we can equate the exponents and solve for $x$. $\begin{aligned} -3x+26 &=8x-1\\\\ x &= \dfrac{27}{11}\end{aligned}$ The answer The answer is $x=\dfrac{27}{11}$. You can check this answer by substituting $\it{x=\dfrac{27}{11}}$ in the original equation and evaluating both sides.